Ymmärsin sanoituksen tällä kertaa paremmin.

Ymmärsin means I understood. It is the past tense of ymmärtää = to understand.

More specifically:

• ymmär- = the verb stem
• -si- = past tense marker
• -n = I

So the sentence is talking about something the speaker understood in the past.

Finnish often leaves out the subject pronoun when it is clear from the verb ending.

Here, ymmärsin already tells us the subject is I, because the ending -n marks first person singular. So minä is not necessary.

You could say Minä ymmärsin sanoituksen tällä kertaa paremmin, but it would usually sound more emphatic, as if you were stressing I.

The basic form is sanoitus. In this sentence it means the lyrics or the wording/text of a song.

A thing that may feel strange to an English speaker is that Finnish can use a singular word here where English often uses the plural lyrics. So:

• sanoitus = the lyrics of a song, treated as one whole text
• English: lyrics = normally plural

So sanoituksen here means the lyrics.

Here sanoituksen is the object of the verb ymmärsin. Finnish uses the total object when the object is understood as a whole, complete thing.

So:

• Ymmärsin sanoituksen = I understood the lyrics as a whole
• sanoituksen is the total-object form, which in the singular usually looks the same as the genitive singular

A learner may compare it with:

• Ymmärsin sanoitusta = I understood the lyrics partially / some of the lyrics / the lyrics to some extent

So the -n form suggests a more complete, bounded result.

Tällä kertaa means this time or on this occasion.

It is a fixed and very common Finnish expression:

• tällä = a case form of tämä (this)
• kertaa = a form of kerta (time, occasion)

You do not normally say tämä kerta when you mean this time in this kind of sentence. The idiomatic expression is tällä kertaa.

So:

• Tällä kertaa ymmärsin paremmin = This time I understood better

Because paremmin is an adverb, and it modifies the verb ymmärsin.

The sentence is not saying that the lyrics were better. It is saying that the speaker understood them better.

Compare:

• parempi = better as an adjective
• paremmin = better as an adverb

Examples:

• Parempi laulu = a better song
• Laulan paremmin = I sing better

In this sentence:

• Ymmärsin ... paremmin = I understood ... better

No, Finnish word order is fairly flexible. The sentence Ymmärsin sanoituksen tällä kertaa paremmin is natural, but other orders are also possible.

For example:

• Tällä kertaa ymmärsin sanoituksen paremmin.
• Ymmärsin sanoituksen paremmin tällä kertaa.

These all mean roughly the same thing, but the emphasis changes slightly.

The given version sounds natural and balanced:

• first the action: Ymmärsin
• then what was understood: sanoituksen
• then the time frame: tällä kertaa
• then the comparison: paremmin

Often, yes. Sanat also means lyrics/words, and in everyday speech many people might say something like Ymmärsin sanat tällä kertaa paremmin.

The difference is roughly this:

• sanoitus = the lyrics as a text or lyrical wording
• sanat = the words/lyrics

In many contexts both work, but sanoituksen sounds a bit more like the lyrical text as a whole.